November 1, 1887.] 



KNO^A^LEDGE ♦ 



11 



that all the numbers from 1 to 32, and from 225 to 25G— 

 i.f., the sixteen lowest and the sixteen highest — fill the two 

 top and the two bottom horizontal lines, and that the order 



No. 1. Compound Magic Square. 



Constructed by the late Mr. J. B. Bassett, by rules of his own 



invention. October 19, ISfiO. 



Sum of each vertical, horizontal, and diagonal column of the 

 largest square = 2,0o6 ; of the inner = 1,542; and of each of the 

 smallest = 514. Hence the relative value of the sum of each column 

 of the several squares is as 4 . 3 . 1. 



of their succession is similar to the knights' leaps over the 

 che.ss-board. So far the rule is fairly sim|)le, but the 

 arrangement of all the intermediate numbers, so far as I 

 have examined the table, seems quite capricious. Yet it 



No. 2 Compound Magic Squaue. 

 Constructed by Mr. W. F. Campion, by rules of his own invention 



1887. 



Sum of each vertical, horizontal, and diagonal column of the 

 outer square, 2,056. By the successive removal of the margins the 

 sum of each horizontal, vertical, and diagonal column is reduced by 

 257. Hence the suras are respectively l,79',l, 1,512, 1,285, 1,028, 

 771, and 514. 



must have been done upon some recognised system, or it 

 could not have been done at all. 



I have iailed in my attempts to discover the rules for 

 the construction of No. 2. Mr. Campion is a storekeeper. 

 He was in Adelaide about a month ago, and he informed 

 me that he discovered at the beginning of the present year 

 the rules for constructing all such squares. In proof of this, 

 he has since then sent me a " Jubilee Magic Square," 

 50 X 50. It covers a sheet of paper about four feet square, 

 and is perfect in every lespect. Thus the sum of each 

 column of the outer square is 62,525, with a constant 

 subtrahend of 2,501 on approaching the centre. The sub- 

 joined was published in " The Riddler " a few weeks ago : — - 



Mr. W. F. Campion has furnished us with several more curious 

 magic squares, including the following : — 



Chinese Magic Square, 9x9. 



In this instance the sum of each vertical, horizontal, and diagonal 

 column of the ulterior square of 3x3 is 123; (hat of the ne.^t, 

 namely, 5 x 5, is 205 ; th.at of the ne.xt, namely, 7x7, 287 ; and tliat 

 of the outermost 369 ; thus the sum of every column is a multiple 

 of 41. 



NOTE ON EUCLID (I. 32). 



(Prop. 3, Bk. I.; Axio.m 12, Bk. I.; and Parallels). 



\ COERESPONDENT asks whether Euclid 

 I. 32 cannot be proved by mechanical con- 

 .siderations independently of axiom 12. 



It has always seemed to me that the pro- 

 position is self-evident, the mind at once 

 picturing some such angular measurement 

 as is .shown in fig. 1. The arrowed rod pq, 

 pivoting round A to position jiq, measures the angle A ; 

 then, pivoting round p, to position p'q', mea^-ures the angle n ; 



Fig. 1. 



and lastly, pivoting round c to position p'q', measures the 

 angle c. It has manifestly pivoted through two right 



